Identification of Time Varying Cardiac Disease
State Using a Minimal Cardiac Model with Reflex
Actions
14th IFAC SYMPOSIUM ON SYSTEM IDENTIFICATION, SYSID-2006
C. E. Hann1, S. Andreassen2, B. W. Smith2, G. M. Shaw3, J. G. Chase1,
P. L. Jensen4
1Department
of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand
for Model-based Medical Decision Support, Aalborg University, Aalborg, Denmark
3 Department of Intensive Care Medicine, Christchurch Hospital, Christchurch, New Zealand
4 Department of Cardiology, Aalborg Hospital, Denmark
2Centre
Diagnosis and Treatment
• Cardiac disturbances difficult to diagnose
- Limited data
- Reflex actions
• Minimal Cardiac Model
- Interactions of simple models
- primary parameters
- common ICU measurements
• Increased resistance in pulmonary artery – pulmonary
embolism, atherosclerotic heart disease
• Require fast parameter ID
Heart Model
D.E.’s and PV diagram
V  Q1  Q2
P  P  Q1 R1
Q1  1 2
L1
P  P3  Q2 R2
Q 2  2
L2
P2  e(t ) Ees (V  Vd )  (1  e(t )) P0 (e (V V0 )  1),
e(t )  e 80( t 0.375)
2
Reflex actions
•
•
•
•
Vaso-constriction - contract veins
Venous constriction – increase
venous dead space
Increased HR
Increased ventricular contractility
Varying HR as a function of Pao
Disease States
•
Pericardial Tamponade
- build up of fluid in pericardium
- dead space volume V0,pcd by 10 ml / 10 heart beats
•
Pulmonary Embolism - Rpul 20% each time
•
Cardiogenic shock
- e.g. blocked coronary artery
- not enough oxygen to myocardium
- Ees,lvf, P0,lvf
•
Septic shock
- blood poisoning
- reduce systemic resistance
•
Hypovolemic shock – severe drop in total blood volume
Healthy Human Baseline
• Healthy human
Output
Value
Volume in left ventricle
111.7/45.7 ml
Volume in right ventricle
112.2/46.1 ml
Cardiac output
5.3 L/min
Max Plv
119.2 mmHg
Max Prv
26.2 mmHg
Pressure in aorta
116.6/79.1 mmHg
Pressure in pulmonary artery
25.7/7.8 mmHg
Avg pressure in pulmonary vein
2.0 mmHg
Avg pressure in vena cava
2.0 mmHg
Model Simulation Results
•
Pericardial tamponade
•
Pulmonary Embolism
Ppu – 7.9 mmHg
CO – 4.1 L/min
MAP – 88.0 mmHg
• All other disease states similarly capture physiological trends and
magnitudes
Add Noise to Identify
• Add 10% uniform distributed noise to outputs to identify
• Apply integral-based optimization
Integral Method - Concept
x  ax  b sin( t )  c, x (0)  1
a  0.5, b  0.2, c  0.8
•
Discretised solution analogous to
measured data
(simple example with
analytical solution )
• Work backwards and find a,b,c
• Current method – solve D. E. numerically or
analytically
1.85
x (t ) 
1.8
 (ab cos t  ba 2 sin t  ca 2  c ))
1.75
1.7
- Find best least squares fit of x(t) to the data
1.65
x
- Non-linear, non-convex optimization,
computationally intense
1.6
q
1.55
1.5
P1  P2
R
• integral method
– reformulate in terms of integrals
1.45
1.4
12
1
( eat ( a  c  ab  ca 2  a 3 )
(a  1)a
2
– linear, convex optimization, minimal computation
13
14
15
16
time
17
18
19
Integral Method - Concept
Integrate x  ax  b sin( t )  c, both sides from t0 to t ( t0  4 )
•
t x dt  t (ax  b sin( t )  c) dt
t
t
0
0
 x(t )  x(t0 )  a t x dt  b t sin( t ) dt  c t 1 dt

•
t
t
t
0
0
0
x(t )  x(t0 )  a t x dt  b(cos( t0 )  cos(t ))  c(t  t0 )
t
0
Choose 10 values of t, between t0  4 and 6 form 10 equations in
3 unknowns a,b,c
a tt x dt  b(1  cos( ti ))  c(ti  t0 )  x (ti )  x (t0 ), i  1,,10
0
Integral Method - Concept
 tt x dt cos( t0 )  cos( t1 ) t1  t0  a   x (t1 )  x (t0 ) 

  

 

  b   


 t
  

x
dt
cos(
t
)

cos(
t
)
t

t
c
x
(
t
)

x
(
t
)

0
10
10
0  
0 
 10
t
1
0
10
0
•
Linear least squares (unique solution)
Method
Starting point
CPU time (seconds)
Solution
Integral
-
0.003
[-0.5002, -0.2000, 0.8003]
Non-linear
[-1, 1, 1]
4.6
[-0.52, -0.20, 0.83]
Non-linear
[1, 1, 1]
20.8
[0.75, 0.32, -0.91]
•
Integral method is at least 1000-10,000 times faster depending on starting point
•
Thus very suitable for clinical application
Identifying Disease State using
All Variables - Simulated
•
•
•
Capture disease states, assume Ppa, Pao, Vlv_max, Vlv_min, chamber flows.
Pericardial tamponade (determining V0,pcd)
Change
True value
(ml)
Optimized
Value
Error (%)
First
180
176
2.22
Second
160
158
1.25
Third
140
138
1.43
Fourth
120
117
2.50
Fifth
100
100
0
Pulmonary Embolism (determining Rpul)
Change
True value
(mmHg s ml-1)
Optimized Value
Error (%)
First
0.1862
0.1907
2.41
Second
0.2173
0.2050
5.67
Third
0.2483
0.2694
8.50
Fourth
0.2794
0.2721
2.60
Fifth
0.3104
0.2962
4.59
Identifying Disease State using
All Variables - Simulated
•
•
Cardiogenic shock (determining [Ees,lvf, P0,lvf] (mmHg ml-1, mmHg)
Change
True values
Optimized
Value
Error (%)
First
[2.59,0.16]
[2.61,0.15]
[0.89,5.49]
Second
[2.30,0.19]
[2.30,0.18]
[0.34,4.39]
Third
[2.02,0.23]
[2.02,0.21]
[0.43,8.03]
Fourth
[1.73,0.26]
[1.70,0.24]
[1.48,9.85]
Fifth
[1.44,0.30]
[1.43,0.27]
[0.47,9.39]
Septic Shock (determining Rsys)
Change
True value
(mmHg s ml-1)
Optimized
Value
Error (%)
First
1.0236
1.0278
0.41
Second
0.9582
0.9714
1.37
Third
0.8929
0.8596
3.73
Fourth
0.8276
0.8316
0.49
Fifth
0.7622
0.7993
4.86
Identifying Disease State using
All Variables - Simulated
•
Hypovolemic Shock (determining stressed blood volume)
Change
True value
(ml)
Optimized Value
Error (%)
First
1299.9
1206.5
7.18
Second
1177.3
1103.7
6.26
Third
1063.1
953.8
10.28 (hmm!)
Fourth
967.8
1018.9
5.28
Fifth
928.5
853.4
8.10
Preliminary Animal Model
Results
• Pulmonary embolism induced in pig (collaborators lab in Belgium)
• Identifying changes in pulmonary resistance, Rpul
Left Ventricle
•
•
•
•
8 heartbeats
Essential dynamics captured
Remaining issues with sensor locations etc
Aortic stenosis as well?
Conclusions
• Minimal cardiac model  simulate time varying disease states
• Accurately captured physiological trends and magnitudes
 capture wide range of dynamics
• Integral based parameter ID method
- errors from 0-10%, with 10% noise
- identifiable using common measurements
• Rapid feedback to medical staff
Acknowledgements
Engineers and Docs
Dr Geoff Chase
Dr Geoff Shaw
The Danes
Steen
Andreassen
The honorary Danes
Dr Bram Smith
Questions ???